arX

iv:1

405.

0216

v1 [

cond

-mat

.mtr

l-sc

i] 1

May

201

4 Cellular buckling in stiffened plates∗

M. Ahmer Wadee and Maryam Farsi

Department of Civil & Environmental Engineering,

Imperial College London, London SW7 2AZ, UK

Abstract

An analytical model based on variational principles for a thin-walled stiffenedplate subjected to axial compression is presented. A system of nonlinear differen-tial and integral equations is derived and solved using numerical continuation. Theresults show that the system is susceptible to highly unstable local–global mode in-teraction after an initial instability is triggered. Moreover, snap-backs in the responseshowing sequential destabilization and restabilization, known as cellular buckling orsnaking, arise. The analytical model is compared to static finite element modelsfor joint conditions between the stiffener and the main plate that have significantrotational restraint. However, it is known from previous studies that the behaviour,where the same joint is insignificantly restrained rotationally, is captured better byan analytical approach than by standard finite element methods; the latter beingunable to capture cellular buckling behaviour even though the phenomenon is clearlyobserved in laboratory experiments.

1 Introduction

The buckling of thin plates that are stiffened in the longitudinal direction represents a struc-tural instability problem of enormous practical significance (Murray, 1973; Ronalds, 1989;Sheikh et al., 2002; Ghavami & Khedmati, 2006). Since they are primarily used in struc-tures where mass efficiency is a key design consideration, stiffened plates tend to compriseslender plate elements that are vulnerable to a variety of different elastic instability phe-nomena. In the current work, the classic problem of a panel comprising a uniform flat platestiffened by several evenly spaced blade-type longitudinal stiffeners under axial compres-sion, made from a linear elastic material, is studied using an analytical approach. Underthis type of loading, wide panels with many stiffeners can be divided into several struts,each with a single stiffener. These individual struts are primarily susceptible to a global (oroverall) mode of instability namely Euler buckling, where flexure about the axis parallel

∗Accepted for publication in Proc. R. Soc. A

1

to the main plate occurs once the theoretical global buckling load is reached. However,when the individual plate elements of the strut cross-section, namely the main plate andthe stiffener, are relatively thin or slender, elastic local buckling of these may also occur; ifthis happens in combination with the global instability, the resulting behaviour is usuallyfar more unstable than when the modes are triggered individually (Budiansky, 1976).

In the current work, the development of a variational model is presented that accountsfor the interaction between the global Euler buckling mode and the local buckling modeof the stiffener, such that the perfect and imperfect elastic post-buckling response canbe evaluated. A system of nonlinear ordinary differential equations subject to integralconstraints is derived and solved using the numerical continuation package Auto-07p

(Doedel & Oldeman, 2011). It is indeed found that the system is highly unstable wheninteractive buckling is triggered; snap-backs in the response show a sequence of destabi-lization and restabilization derived from the mode interaction and the stretching of theplate surface during local buckling respectively. This results in a progressive spreading ofthe initial localized buckling mode. This latter type of response has become known in theliterature as cellular buckling (Hunt et al., 2000) or snaking (Burke & Knobloch, 2007) andit is shown to appear naturally in the current numerical results. The effect is particularlystrong where the rotational restraint provided at the joint between the main plate and thestiffener is negligible. As far as the authors are aware, this is the first time this phenomenonhas been found analytically in stiffened plates undergoing global and local buckling simul-taneously. Similar behaviour has been discovered in various other mechanical systems suchas in the post-buckling of cylindrical shells (Hunt et al., 2003), the sequential folding ofgeological layers (Wadee & Edmunds, 2005), the buckling of thin-walled I-section beams(Wadee & Gardner, 2012) and columns (Wadee & Bai, 2013).

In previous work, cellular buckling has been captured in physical tests for some closelyrelated structures that suffer from local–global mode interaction (Becque & Rasmussen, 2009;Wadee & Gardner, 2012); it is revealed by the buckling elements exhibiting a progressivelyvarying deformation wavelength. However, by increasing the rotational stiffness of theaforementioned joint, the snap-backs are moderated and the equilibrium path shows aninitially smoother response with the snap-backs appearing later in the interactive bucklingprocess. For these cases, the results from a purely numerical model, formulated within thecommercial finite element (FE) package Abaqus (2011), are used to validate the analyticalmodel and exhibit very good correlation, leading to the conclusion that the physics of thesystem is captured accurately by the analytical model.

2 Analytical Model

2.1 Modal Description

Consider a thin-walled simply-supported plated panel that has uniformly spaced stiffenersabove and below the main plate, as shown in Figure 1 with the associated coordinatesystem. The panel length is L and the spacing between the stiffeners is b. It is made from

2

Simply-supported stiffened panel:

b

Focus on this portion

L

(a)

z L

yP Neutral axis of bending

(b)

ts

tstp

h1

h2

y

y

bx

(c)

zy W(z)

Sway mode:

(d)

zy θ(z)

Tilt mode:

(e)

Figure 1: (a) Axially compressed simply-supported stiffened panel of length L. (b) Ele-vation and (c) cross-section of the representative panel portion compressed by a force Papplied to the centroid. (d–e) Sway and tilt components respectively of the major axisglobal buckling mode.

3

a linear elastic, homogeneous and isotropic material with Young’s modulus E, Poisson’sratio ν and shear modulus G = E/[2(1+ν)]. A portion of the panel that is representative ofits entirety can be isolated as a strut, shown in Figure 1(b–c), since the transverse bendingcurvature of the panel during buckling would be relatively small, particularly if L ≪ nsb,where ns is the number of stiffeners in the panel. The main plate (width b and thicknesstp) has upper and lower stiffeners (heights h1 and h2 respectively with equal thickness ts)connected to it. The strut is loaded by an axial force P that is applied to the centroidof the cross-section. Presently, however, although the model is formulated for the generalcase, the numerical results focus on the practically significant case where the stiffeners areonly connected to one side of the panel, i.e. where h2 = tp/2.

Moreover, the geometries are chosen such that global (Euler) buckling about the x-axisis the first instability mode encountered by the strut, before any local buckling in the mainplate or stiffener occurs. The formulation for global buckling is based on small deflectionassumptions, since it is well known that it has an approximately flat (or weakly sta-ble) post-buckling response (Thompson & Hunt, 1973). The study is primarily concernedwith global buckling forcing the stiffener to buckle locally shortly after or even simulta-neously. It has been shown in several works (Hunt & Wadee, 1998; Wadee et al., 2010;Wadee & Gardner, 2012; Wadee & Bai, 2013) that shear strains need to be included tomodel the local–global mode interaction analytically. For thin-walled components, ratherthan, for instance, soft core materials used in sandwich structures (Wadee et al., 2010),Timoshenko beam theory gives sufficiently accurate results (Wadee & Gardner, 2012). Thiscan be applied to global flexural buckling within the framework of two degrees of freedom,known as “sway” and “tilt” in the literature (Hunt & Wadee, 1998). These are associatedwith functions describing the lateral displacement W and the angle of inclination θ respec-tively defining the appropriate kinematics, as shown in Figure 1(c). From linear theory, itcan be shown that W (z) and θ(z) can be represented by the following expressions:

W (z) = −qsL sinπz

L, θ(z) = qtπ cos

πz

L. (1)

The quantities qs and qt are the generalized coordinates of the sway and tilt componentsof global buckling respectively. The corresponding shear strain γyz during bending is givenby the following expression:

γyz =dW

dz+ θ = − (qs − qt)π cos

πz

L. (2)

Of course, if standard Euler–Bernoulli bending were used then γyz would be zero and qswould be equal to qt. The initial global buckling displacement is assumed to put the stiff-ener into extra compression and it therefore becomes vulnerable to local buckling. Thedirection of global buckling is crucially important; if the stiffener goes into extra compres-sion and buckles locally it can lead to excessive deflection. This can induce plasticity withinthe stiffener, leading to the tripping phenomenon that can lead to catastrophic failure ofpanels (Ronalds, 1989; Butler et al., 2001); currently, however, linear elasticity is assumedthroughout.

4

The stiffener is taken to be a flat plate (blade-type) and the local kinematics of itrequire careful consideration. A linear distribution in y for the local in-plane displacementu(y, z) is assumed owing to the application of Timoshenko beam theory. The tips of thestiffeners have free edges but a rotational spring with stiffness cp is included to model theresistance to rotation about the z-axis that the joint provides between the stiffeners andthe main plate. Hence, if cp = 0, the stiffeners are effectively pinned with the main plate.In the generic case (cp > 0), however, the assumed out-of-plane displacement along thewidth of the stiffener w(y, z) can be approximated by a function that is a summation oftrigonometric and polynomial terms. Figure 2 shows the form of the functions w and u

y

w(y,z)

w0(y,z)

y

x

(a)

cp

x

(b)

u(h1-y,0) u(h1-y,L)y

(c)

ε0

ε

z

W0(z)

W(z)z

y θθ0

z

Stress relieved

Flexural rigidity EIp

χχ0

M

ut = y (θ - θ0)

εz,overall = ∂ut

∂zθ

dθ

M

M

dW

(d)

Figure 2: (a) Local out-of-plane displacement w(y, z) and an additional displacementw0(y, z) that represents an initial imperfection in the stiffener geometry; (b) a pinnedconnection with an additional rotational spring of stiffness cp to model joint fixity; (c) thelocal in-plane displacement u(y, z). (d) Introduction of the global imperfection functionsW0 and θ0; for the consideration of the local imperfection, w0 would replace W0 and applyonly to the stiffener.

with the algebraic expressions being given thus:

u(y, z) = Y (y)u(z), w(y, z) = f(y)w(z), (3)

where Y (y) = (y + y)/h1 and

f(y) = B0 +B1Y +B2Y2 +B3Y

3 +B4 sin (πY ) , (4)

5

with B0, B1, B2, B3 and B4 being constants evaluated by solving for the appropriateboundary conditions for the stiffener. The choice of shape for f reflects the possibility ofthe slope at the junction between the main plate and the stiffener being small in the lesscompressed zone of the stiffener. For y = −y, at the junction between stiffener and themain plate, the conditions are:

w(y, z) = 0, D∂2

∂y2w(y, z) = cp

∂

∂yw(y, z), (5)

and for y = h1 − y, the free-end, the conditions are:

D∂2

∂y2w(h1 − y, z) = 0, D

∂3

∂y3w(h1 − y, z) = 0, (6)

where y is the location of the neutral-axis of bending measured from the centre line of themain plate, expressed thus:

y =ts [h

21 − h22]

2 [(b− ts) tp + (h1 + h2) ts]. (7)

By applying the boundary conditions, the function for the deflected shape of the stiffenerw(y, z) is found to be:

w(y, z) =

{

Y − S4π3

6

[

2Y − 3Y 2 + Y 3 −6

π3sin (πY )

]}

w(z), (8)

where:S4 =

{

π[

(Dπ2)/(cph1) + π2/3− 1]}

−1. (9)

In the current study, both perfect and imperfect panels are considered. A displacementW0 corresponding toW and a rotation θ0 corresponding to θ are introduced; the respectiveexpressions being:

W0(z) = −qs0L sinπz

L, θ0 = qt0π cos

πz

L, (10)

with qs0 and qt0 being the amplitudes defining the initial global imperfection. A localimperfection w0(y, z) = f(y)w0(z), see Figure 2(a), is also introduced, the form of whichis derived from a first order approximation of a multiple scale perturbation analysis of astrut on a softening foundation (Wadee et al., 1997):

w0(z) = A0 sech

[

α (z − η)

L

]

cos

[

βπ (z − η)

L

]

, (11)

where z = [0, L] and w0 is symmetric about z = η. This form for w0 has been shown to berepresentative for local–global mode interaction problems in the literature (Wadee, 2000).It enables the study of periodic and localized imperfections, the latter being shown to berelatively more severe for related problems (Wadee, 2002; Wadee & Simoes da Silva, 2005).

6

2.2 Total Potential Energy

The governing equations of equilibrium are derived from variational principles by minimiz-ing the total potential energy V of the system. The total potential energy comprises thecontributions of global and local strain energies of bending, Ubo and Ubl respectively, thestrain energy stored in the “membrane” of the stiffener Um arising from axial and shearstresses, and the work done by the external load PE . The effects of the initial imperfec-tions are evaluated by assuming that the imperfect forms, given by the functions W0 andw0, are stress-relieved. This implies that the elemental moment M and hence the bendingenergy both drop to zero in the unloaded state, see Figure 2(d). The global bending energyaccounts for the main plate and is hence given by:

Ubo =1

2EIp

∫ L

0

(

W − W0

)2

dz =1

2EIp

∫ L

0

(qs − qs0)2 π

4

L2sin2 πz

Ldz, (12)

where dots represent differentiation with respect to z and Ip = (b− ts)t3p/12 + (b− ts)tpy

2

is the second moment of area of the main plate about the global x-axis. The local bendingenergy Ubl, accounting for the stiffener only, is given thus:

Ubl =D

2

∫ L

0

∫ h1−y

−y

{

[

∂2(w − w0)

∂z2+∂2(w − w0)

∂y2

]2

−2 (1− ν)

[

∂2(w − w0)

∂z2∂2(w − w0)

∂y2−

(

∂2(w − w0)

∂z∂y

)2]}

dy dz,

=D

2

∫ L

0

[

{f 2}y (w − w0)2 +

{

f ′′2}

y(w − w0)

2 + 2ν{

ff ′′}

y(w − w0)(w − w0)

+2(1− ν){

f ′2}

y(w − w0)

2]

dz,

(13)

where primes denote differentiation with respect to y, D is the plate flexural rigidity givenby Et3s/[12(1− ν2)] and:

{F (y)}y =

∫ h1−y

−y

F (y) dy, (14)

where F is an example function. The compressive side of the panel only contributes tothe local bending energy once global buckling occurs. The membrane energy Um is derivedfrom considering the direct strains (ε) and shear strains (γ) in the stiffener. The globalbuckling distribution for the longitudinal strain εz can be obtained from the tilt componentof displacement from the global mode, thus:

εz,global = yd

dz(θ − θ0) = −y (qt − qt0)

π2

Lsin

πz

L. (15)

The local mode contribution is based on von Karman plate theory. A purely in-planecompressive strain ∆ is also included. The direct strain for the main plate is given by:

7

εzp = −∆. The combined expressions for the direct strains for the top and bottom stiffenersεzt and εzb respectively, including local and global buckling are:

εzt = −y (qt − qt0)π2

Lsin

πz

L−∆+

∂u

∂z+

1

2

(

∂w

∂z

)2

−1

2

(

∂w0

∂z

)2

,

= −y (qt − qt0)π2

Lsin

πz

L−∆+

(

y + y

h1

)

u+1

2{f 2}y

(

w2 − w20

)

,

(16)

εzb = −y (qt − qt0)π2

Lsin

πz

L−∆. (17)

The complete expression for the membrane strain energy Um is given by:

Um = Ud + Us

=1

2

∫ L

0

∫ ts/2

−ts/2

[∫ h1−y

−y

(

Eε2zt +Gγ2yzt)

dy +

∫

−y

−(h2+y)

(

Eε2zb +Gγ2yzb)

dy

]

dx dz,(18)

where Ud is the contribution from direct strains, which is given by the terms multipliedby the Young’s modulus E; whereas Us, the contribution arising from the shear strains,is given by the terms multiplied by the shear modulus G. The transverse componentof the strain εy is neglected since it has been shown that it has no effect on the post-buckling stiffness of a long plate with three simply-supported edges and one free edge(Koiter & Pignataro, 1976). The total direct strain energy Ud is therefore:

Ud =1

2Ets

∫ L

0

{

1

3

[

(h1 − y)3 + (h2 + y)3]

(qt − qt0)2 π

4

L2sin2 πz

L

+∆2 (h1 + h2) +[

(h1 − y)2 − (h2 + y)2]

∆(qt − qt0)π2

Lsin

πz

L

+ h1

[

1

3u2 +

1

4h1{f 4}y

(

w2 − w20

)2+

{

Y f 2

h1

}

y

u(

w2 − w20

)

]

− (qt − qt0)h1π

2

Lsin

πz

L

[(

2

3h1 − y

)

u+1

h1{yf 2}y

(

w2 − w20

)

]

− h1∆

[

u+1

h1{f 2}y

(

w2 − w20

)

]

+

(

tpts

)

(b− ts)∆2

}

dz.

(19)

The shear strain energy Us requires the shear strain γyz, which is also modelled separatelyfor the compression and the tension side of the stiffeners. The general expression for theshear strain γyzi in the stiffeners is thus:

γyzi =d

dz(W −W0) + (θ − θ0) +

∂u

∂y+∂w

∂z

∂w

∂y−∂w0

∂z

∂w0

∂y, (20)

and the expressions for the top and bottom stiffeners are given respectively:

γyzt = − [(qs − qs0)− (qt − qt0)] π cosπz

L+

u

h1+{

ff ′}

y(ww − w0w0),

γyzb = − [(qs − qs0)− (qt − qt0)] π cosπz

L,

(21)

8

hence, the expression for Us is:

Us =1

2Gts

∫ L

0

{

[(qs − qs0)− (qt − qt0)]2 π2 cos2

πz

L(h1 + h2)

+1

h1

[

u2 + h1{

(ff ′)2}

y(ww − w0w0)

2 + 2{

ff ′}

yu (ww − w0w0)

]

− [(qs − qs0)− (qt − qt0)]

[

2u+ 2{

ff ′}

y(ww − w0w0)

]

π cosπz

L

}

dz.

(22)

The strain energy stored in the rotational spring connecting the stiffeners to the main plateis:

Usp =1

2cp

∫ L

0

{

∂

∂y[w(−y)− w0(−y)]

}2

dz =1

2cp

∫ L

0

[f ′2(−y)](w − w0)2 dz, (23)

where F (−y) means that the example function F is evaluated at y = −y. The finalcomponent of V is the work done by the axial load P , which is given by:

PE =P

2

∫ L

0

[

2∆ + q2sπ2 cos2

πz

L− 2

(

h2 + y

h1 + h2

)

u

]

dz, (24)

where the total end-displacement E comprises components from pure squash and swayfrom global buckling combined with the local buckling of the stiffener respectively. Thetotal potential energy V is given by the summation of all the strain energy terms minusthe work done, thus:

V = Ubo + Ubl + Um + Usp − PE . (25)

2.3 Variational Formulation

The governing equilibrium equations are obtained by performing the calculus of variationson the total potential energy V following a well established procedure that has been detailedin Hunt and Wadee (1998). The integrand of the total potential energy V can be expressedas the Lagrangian (L) of the form:

V =

∫ L

0

L (w, w, w, u, u, z) dz. (26)

The first variation of V is given by:

δV =

∫ L

0

[

∂L

∂wδw +

∂L

∂wδw +

∂L

∂wδw +

∂L

∂uδu+

∂L

∂uδu

]

dz, (27)

to determine the equilibrium states, V must be stationary, hence the first variation δV mustvanish for any small change in w and u. Since δw = d(δw)/ dz, δw = d(δw)/dz and sim-ilarly δu = d(δu)/dz, integration by parts allows the development of the Euler–Lagrange

9

equations for w and u; these comprise a fourth-order and a second-order nonlinear differ-ential equation for w and u respectively. To facilitate the solution of the equations withinthe package Auto-07p, the variables are rescaled with respect to the non-dimensionalspatial coordinate z, defined as z = 2z/L. Similarly, the non-dimensional out-of-plane andin-plane displacements w and u are defined with the scalings 2w/L and 2u/L respectively.Note that the scalings exploit symmetry about the midspan and the equations are hencesolved for half the strut length; this assumption has been shown to be acceptable for caseswhere global buckling is critical (Wadee, 2000). The non-dimensional differential equationsare thus:

˜....w − ˜....w0 +L2

2{f 2}y

[

ν {ff ′′}y − (1− ν){

f ′2}

y

]

(

˜w − ˜w0

)

+ k (w − w0)

− D

[

{f 4}y{f 2}y

(

3 ˜w2 ˜w − ˜ww20 − 2 ˜w0

˜w0˜w)

+{2Y f 2}y{f 2}y

(

˜u ˜w + ˜w ˜u)

−2∆ ˜w − 2 (qt − qt0)π2

L

{yf 2}y{f 2}y

(

sinπz

2˜w +

π

2cos

πz

2˜w

)

]

−GL2w

2 {f 2}y

[

{

(ff ′)2}

y

(

˜w2 + w ˜w − ˜w20 − w0

˜w0

)

+1

h1{ff ′}y

˜u

+ [(qs − qs0)− (qt − qt0)]π2

L{ff ′}y sin

πz

2

]

= 0,

(28)

˜u−3

4

G

Dψ

{

ψ[

u+ {ff ′}y(

w ˜w − w0˜w0

)

]

− 2π [(qs − qs0)− (qt − qt0)] cosπz

2

}

−

{

3Y

h1f 2

}

y

(

˜w ˜w + ˜w0˜w0

)

+1

2(qt − qt0)π

3

(

ψ −3y

2L

)

cosπz

2= 0,

(29)

where the rescaled quantities are:

D = EtsL2/8D, G = GtsL

2/8D, k =L4

16 {f 2}y

[

{

f ′′2}

y+ cpf

′2(−y)/D

]

, (30)

with w0 = 2w0/L, ψ = L/h1 and f ′2(−y) is as described previously. Equilibrium alsorequires the minimization of the total potential energy with respect to the generalizedcoordinates ∆, qs and qt leading to the three integral equations in nondimensional form:

∂V

∂∆= ∆

[

1 +h2h1

+tp(b− ts)

ts

]

−P

Etsh1+ (qt − qt0)

π

h1L

[

(h1 − y)2 − (h2 + y)2]

−1

4

∫ 2

0

[

˜u+1

h1

{

f 2}

y

(

˜w2 − w20

)

]

dz = 0, (31)

∂V

∂qs= π2 (qs − qs0) + s [(qs − qs0)− (qt − qt0)]−

PL2

EIpqs

−sφ

2π

∫ 2

0

cosπz

2

[

u+ {ff ′}y(

w ˜w − w0˜w0

)

]

dz = 0, (32)

10

∂V

∂qt= π2 (qt − qt0) + Γ3∆− t [(qs − qs0)− (qt − qt0)]−

1

2

∫ 2

0

{

sinπz

2

[

Γ1˜u

+Γ2

(

˜w2 − ˜w20

)

]

−tφ

πcos

πz

L

[

u+ {ff ′}y(

w ˜w − w0˜w0

)

]

}

dz = 0, (33)

where the rescaled quantities are:

Γ1 =Lh1 (2h1 − 3y)

(h1 − y)3 + (h2 + y)3, Γ2 =

3L {yf 2}y

(h1 − y)3 + (h2 + y)3,

Γ3 =6L

[

(h1 − y)2 − (h2 + y)2]

π[

(h1 − y)3 + (h2 + y)3] , φ =

L

h1 + h2,

s =Gts(h1 + h2)L

2

EIp, t =

3GL2(h1 + h2)

E[

(h1 − y)3 + (h2 + y)3] .

(34)

Since the strut is an integral member, Equations (32)–(33) provide a relationship linkingqs and qt before any interactive buckling occurs, i.e. when w = u = 0. This link is assumedto hold also between qs0 and qt0, which acts as a simplification by reducing the number ofglobal imperfection amplitude parameters to one; this relationship is given by:

qs0 =(

1 + π2/t)

qt0. (35)

The boundary conditions for w and u and their derivatives are for simply-supported con-ditions at z = 0 and for symmetry conditions at z = 1:

w(0) = ˜w(0) = ˜w(1) =...w(1) = u(1) = 0, (36)

with a further condition from matching the in-plane strain:

1

3˜u(0) +

1

2

{

Y

h1f 2

}

y

[

˜w2(0)− w20(0)

]

−1

2∆ +

P

Etsh1

(

h2 + y

h1 + h2

)

= 0. (37)

It is worth noting that the terms in braces in Equations (28)–(37) are integrated withrespect to the original definition of y.

Linear eigenvalue analysis for the perfect strut (qs0 = qt0 = A0 = 0) is conducted todetermine the critical load for global buckling PC

o . By considering that the Hessian matrixVij, thus:

Vij =

[

∂2V∂q2

s

∂2V∂qs∂qt

∂2V∂qt∂qs

∂2V∂q2

t

]

(38)

at the critical load is singular, in conjunction with the pre-buckling condition qs = qt =w = u = 0, the critical load for global buckling is obtained:

PCo =

π2EIpL2

[

1 +s

π2 + t

]

. (39)

If the limit G→ ∞ is taken, a primary assumption in Euler–Bernoulli bending theory, PCo

reduces to the expected classical Euler column buckling load.

11

3 Numerical study

The full system of equilibrium equations are difficult to solve analytically. The continua-tion and bifurcation software Auto-07p is thus used; it has been shown in previous work(Wadee & Gardner, 2012) to be an ideal tool to solve the equations for this kind of me-chanical system. The solver is adept at locating bifurcation points and tracing branchingpaths as model parameters are varied. To demonstrate this, an example set of geomet-ric and material properties are chosen thus: L = 5000 mm, b = 120 mm, tp = 2.4 mm,ts = 1.2 mm, h1 = 38 mm, h2 = tp/2, E = 210 kN/mm2, ν = 0.3. The global criticalload PC

o can be calculated using Equation (39), whereas the local buckling critical stressσCl can be evaluated using the well-known formula σC

l = kpDπ2/(b2t), where the coefficient

kp depends on plate boundary conditions; limiting values being kp = 0.426 and kp = 1.247for a long stiffener connected to the main plate with one edge free and the edge definingthe junction between the stiffener and the main plate being taken to be pinned or fixedrespectively. For the panel selected, Table 1 shows that global buckling is critical and that

σCo (N/mm2) σC

l,s (N/mm2) σCl,p (N/mm2) Critical mode

4.89 80.22 373.28 Global

Table 1: Theoretical values of the global and local critical buckling stresses (σCo and σC

l )respectively; subscripts “p” and “s” refer to the main plate and the stiffener respectively.The expression for σC

o = PCo /A, where A is the cross-sectional area of the strut.

the stiffener could be the next in line to buckle locally since its critical stress is much lessthan that of the main plate.

3.1 Analytical model results

To solve the governing equations, the principal parameters used in the numerical contin-uation procedure in Auto-07p were interchangeable although generally qs was varied forcomputing the equilibrium paths for the distinct buckling modes. However, the load Pwas used as the principal continuation parameter for computing the interactive bucklingpaths. The perfect post-buckling path was computed first from PC

o and then a sequenceof bifurcation points were detected on the weakly stable post-buckling path; the value ofqs closest to the critical bifurcation point (C) was identified as the secondary bifurcationpoint (S), see Figure 3, and denoted as qSs . This triggered interactive buckling and a newequilibrium path was computed. Typically this reduced the load P with w and u becomingnon-trivial as the interaction advanced. On the interactive buckling path, a progressivesequence of destabilization and restabilization, exhibited as a series of snap-backs in theequilibrium paths was observed – the signature of cellular buckling (Hunt et al., 2000;Wadee & Edmunds, 2005; Wadee & Gardner, 2012; Wadee & Bai, 2013). In Figure 3, thefirst such snap-back is labelled as T with qs = qTs and the existence of it marks the begin-ning of the rapid spreading of the buckling profile from being localized to being eventuallyperiodic.

12

PC

o

S

P

qs

Run 1

Run 2

C

qS

sq

T

s

T PC

o

P

qsqs0

SCT

Run 1

Perfect case Imperfect case

Figure 3: Diagrammatic representation of the sequence for computing the equilibriumpaths for the (left) perfect and (right) imperfect cases.

For the example panel being considered, Figure 4 shows equilibrium plots of the nor-malized axial load p = P/PC

o versus the normalized buckling amplitudes of (a) the globalmode of the strut and (b) the local mode of the stiffener. The graph in (c) shows therelative amplitudes of the global and local buckling modes. Finally, (d) shows the rela-tionship between the sway and tilt amplitudes of global buckling, which are almost equal;indicating that the shear strain from the global mode is small but, importantly, not zero.Figure 5 shows a selection of 3-dimensional representations of the deflected strut that in-clude the components of global buckling (qs and qt) and local buckling (w and u) for Sand cells C4, C6, and C8 respectively. Figure 6 illustrates the corresponding progressionof the numerical solutions for the local buckling functions w and u. The results for theexample clearly shows cellular buckling with the spiky features in the graphs in Figure4(a–c). Moreover, the buckling patterns, as seen in Figure 5, clearly show an initiallylocalized buckle gradually spreading outwards from the panel midspan with more peaksand troughs.

In the literature (Hunt et al., 2000; Wadee & Edmunds, 2005) the Maxwell load (PM)is calculated for snaking problems, which represents a realistic lower bound strength for thesystem. Presently, however, it is more complex to determine such a quantity because thesystem axial load P does not oscillate about a fixed load PM as the deformation increases.This is primarily owing to the fact that, unlike systems that exhibit cellular buckling,such as cylindrical shells and confined layered structures (Hunt et al., 2000), there aretwo effective loading sources as the mode interaction takes hold: the axial load P , whichgenerally decreases and the sinusoidally varying (in z) tilt generalized coordinate qt, whichrepresents the axial component of the global buckling mode that generally increases. Thenotion of determining the “body force”, discussed in Hunt and Wadee (1998), could beused as way to calculate the Maxwell load, but this has been left for future work.

Hitherto, the results have been presented for the case where the joint between the mainplate and the stiffener is pinned (cp = 0). Figure 7 shows how the introduction of the jointstiffness affects the response. First of all, the value of σC

l,s is increased since rotation atthe joint is more restrained. This, in turn, increases the value of qSs and qTs at different

13

0 0.005 0.01 0.015 0.020

0.2

0.4

0.6

0.8

1

qs

p

S C2C4

C6

C8

(a)

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

wmax/ts

p

S C2C4

C6

C8

(b)

0 0.005 0.01 0.015 0.020

0.5

1

1.5

2

qs

wmax/t s

S

C2

C4

C6

C8

(c)

0 0.005 0.01 0.015 0.020

0.005

0.01

0.015

0.02

qs

q t

S

C2

C4

C6

C8

(d)

Figure 4: Numerical solutions from the analytical model: equilibrium paths. Normalizedforce p versus the normalized amplitudes of (a) global buckling qs and (b) local bucklingof the stiffener wmax/ts. Modal amplitude comparisons: (c) local versus global; (d) swayversus tilt.

14

−50050

0

1000

2000

3000

4000

5000−100

0

100

x

z

y

(a)

−50050

0

1000

2000

3000

4000

5000−100

0

100

x

z

y

(b)

−50050

0

1000

2000

3000

4000

5000−100

0

100

x

z

y

(c)

−50050

0

1000

2000

3000

4000

5000−100

0

100

x

z

y

(d)

Figure 5: Numerical solutions from the analytical model visualized on 3-dimensional repre-sentations of the strut. (a) Secondary bifurcation point S (p = 1.00), (b) cell C4 (p = 0.976),(c) cell C6 (p = 0.926) and (d) cell C8 (p = 0.865). All dimensions are in millimetres, butthe local buckling displacements in the stiffener are scaled by a factor of 5 to aid visual-ization.

rates. It can be seen that as cp is increased the initial interactive mode that emerges fromthe secondary bifurcation S is more localized with a smaller wavelength (Figure 7) andresembles the more advanced cells shown in Figure 6 directly. Moreover, it is demonstratedthat the snap-backs occur later in the interactive buckling process.

4 Validation

The commercial FE software Abaqus was selected to validate the results from Auto-

07p where appropriate. Four-noded shell elements with reduced integration were usedto model the structure. Rotational springs were also used along the length to simulatethe boundary conditions at the joint of the stiffener with the main plate. An eigenvalue

15

0 1000 2000 3000 4000 5000−2

02

w (

mm

)

z (mm)0 1000 2000 3000 4000 5000

−0.020

0.02

u (m

m)

z (mm)

0 1000 2000 3000 4000 5000−2

02

w (

mm

)

z (mm)0 1000 2000 3000 4000 5000

−0.020

0.02

u (m

m)

z (mm)

0 1000 2000 3000 4000 5000−2

02

w (

mm

)

z (mm)0 1000 2000 3000 4000 5000

−0.020

0.02

u (m

m)

z (mm)

0 1000 2000 3000 4000 5000−2

02

w (

mm

)

z (mm)0 1000 2000 3000 4000 5000

−0.020

0.02

u (m

m)

z (mm)

Figure 6: Numerical solutions for the local buckling mode out-of-plane deflection w (left)and in-plane deflection u (right) showing the progressive spreading of the stiffener deflectionfor cells C2, C4, C6 and C8 (top to bottom respectively) of the perfect case from theanalytical model.

analysis was used to calculate the critical buckling loads and eigenmodes. The nonlinearpost-buckling analysis was performed with the static Riks (1979) method with the afore-mentioned eigenmodes being used to introduce the necessary geometric imperfection tofacilitate this. For the numerical example, the strut described in §3 has a rotational springstiffness cp = 1000 Nmm/mm. As discussed at the end of the previous section, this basi-cally increases the gap between triggering the interactive buckling mode and the onset ofcellular buckling behaviour since the initial local buckling mode is naturally more spreadthrough the length; this smoothing and hence simplification of the response potentiallyallows for a better comparison with the analytical model.

Linear buckling analysis shows that global buckling is the first eigenmode and thecorresponding critical load and critical stress are PC

o = 1.64 kN and σCo = 4.95 N/mm2

respectively, which is a difference of approximately 1% from the value calculated fromthe analytical model (Table 1). Figure 8 shows the comparisons between the analyticalresults from Auto-07p and the numerical results from Abaqus. This is shown for thecase with an initial global imperfection (W0) where qs0 = 0.001 is combined with a localimperfection (w0) where A0 = 0.12 mm, α = 8.0, β = 35 and η = L/2, which matchesthe initial imperfection from Abaqus for the analytical model such that a meaningfulcomparison can be made. The graphs in (a–b) show the normalized axial load p versus thebuckling amplitudes of the global mode and the local deflection of the stiffener respectively.The graph in (c) shows the local versus the global buckling modal amplitudes; all threegraphs show excellent correlation in all aspects of the mechanical response.

Figure 8(d) shows the local out-of-plane deflection profiles at the respective locations

16

0 2000 4000−2

02

w(m

m)

0 2000 4000−1

01

w(m

m)

0 2000 4000−1

01

x 10−3

w(m

m)

0 2000 4000−1

01

x 10−3

w(m

m)

z(mm)

(a) p = 0.99

0 2000 4000−2

02

w(m

m)

0 2000 4000−1

0

1

w(m

m)

0 2000 4000−1

0

1

w(m

m)

0 2000 4000−1

0

1

w(m

m)

z(mm)

(b) p = 0.96

0 200 400 600 800 10000

0.01

0.02

0.03

q s

cp (Nmm/mm)

qSsqTs

0 0.01 0.02 0.030

0.2

0.4

0.6

0.8

1

qs

p

increasing cp

(c)

Figure 7: (a–b) Numerical solutions from the analytical model for the local out-of-plane deflection w in the initial interactive buckling stage, where cases for cp =0, 10, 100, 1000 Nmm/mm are shown from top to bottom respectively. (c) Curves showingthe distribution of qSs and qTs , and the load p versus qs for the increasing values of cp givenin (a–b).

(i)–(iii) shown in Figure 8(a–c), the comparisons being for the same values of qs. These,again, show excellent correlation between the results from Auto-07p and Abaqus, partic-ularly near the peak load. A visual comparison between the 3-dimensional representationof the strut from the analytical and the FE models is also presented in Figure 9. The cor-relation with Abaqus, for a relatively high value of cp, is seen to be excellent, but when cpis decreased the analytical model exhibits the characteristic snap-backs earlier (see Figures4 and 7). However, the Abaqus solution for static post-buckling, which relies on assumingan initially imperfect geometry, does not (and indeed possibly cannot) obtain the progres-sive cellular buckling solution. This is not entirely surprising because the wavelength of theAbaqus solution seems to be fixed throughout the loading history. This demonstrates thatfor situations where buckling wavelengths are likely to change, the conventional method foranalysing instability problems with static FE methods, where initial imperfections affineto a combination of eigenmodes are introduced and analysed using a nonlinear solver (e.g.the Riks method), may not be able to capture some experimentally observed physical phe-nomena. As an alternative, the authors attempted to model the problem as a dynamicalsystem within Abaqus such that the snap-backs may be replicated more readily. However,computational issues in terms of slow and unreliable convergence stalled this study and

17

0 0.005 0.01 0.015 0.02 0.025 0.030

0.2

0.4

0.6

0.8

1

qs

p

(i)

(ii)

(iii)

AUTOABAQUS

(a)

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

wmax/ts

p

(i)(ii)

(iii)

AUTOABAQUS

(b)

0 0.005 0.01 0.015 0.02 0.025 0.030

0.5

1

1.5

2

qs

wmax/t s

(i)

(ii)

(iii)

AUTOABAQUS

(c)

0 1000 2000 3000 4000 5000−2

0

2

w(m

m)

z(mm)

(i)

0 1000 2000 3000 4000 5000−2

0

2

w(m

m)

z(mm)

(ii)

0 1000 2000 3000 4000 5000−2

0

2

w(m

m)

z(mm)

(iii)

(d)

Figure 8: Comparison of the results from Auto (solid lines) with Abaqus (dashed lines)solutions. Normalized force p versus amplitudes of (a) the global mode qs (b) the localmode wmax/ts. (c) local versus global mode amplitudes; (d) local buckling out-of-planedeflections w of the stiffener for the points shown in (a–c) defined as (i)–(iii).

18

−50050

0

1000

2000

3000

4000

5000−100

0

100

x

z

y

−50050

0

1000

2000

3000

4000

5000−100

0

100

x

z

y

(a)

−50050

0

1000

2000

3000

4000

5000−100

0

100

x

z

y

−50050

0

1000

2000

3000

4000

5000−100

0

100

x

z

y

(b)

−50050

0

1000

2000

3000

4000

5000−100

0

100

x

z

y

−50050

0

1000

2000

3000

4000

5000−100

0

100

x

z

y

(c)

Figure 9: Numerical solutions from (left) Abaqus and (right) Auto visualized on 3-dimensional representations of the strut at the same load level: (a) p = 0.89, (b) p = 0.84,and (c) p = 0.81 respectively). All dimensions are in millimetres, but the local bucklingdisplacements in the stiffener are scaled by a factor of 5 to aid visualization.

19

it has been left for the future. Nevertheless, a recent study of a closely related structure(Wadee & Bai, 2013) clearly showed that the analytical approach matches the physical re-sponse significantly more accurately than static FE methods, in terms of the load versusdisplacement behaviour, and the observed change in the post-buckling wavelength. Theobservation of precisely the cellular buckling response has been reported for the flangesof thin-walled I-section struts experimentally (Becque, 2008), which is very similar to thecurrent case where h1 = h2 (see Figure 1). Hence, the authors are of the opinion thatsince cellular buckling is an observed physical phenomenon for those related structuralcomponents, the current results from the analytical model are also valid.

5 Concluding remarks

The current work identifies a form of cellular buckling in axially compressed stiffened plates.This arises from a potentially dangerous interaction of local and global modes of buckling,with a characteristic sequence of snap-back instabilities in the mechanical response. Itis also found that the effect of the instabilities can be reduced somewhat by increasingthe rotational stiffness of the joint between the main plate and the stiffeners. However,the mode interaction persists and the local buckling profile still changes wavelength, as hasbeen found in recent related studies (Becque & Rasmussen, 2009; Wadee & Gardner, 2012;Wadee & Bai, 2013). The increased rotational stiffness of the joint allows the model to bevalidated from a purely numerical formulation in FE software. However, earlier studiessuggest that for smaller values of the aforementioned stiffness, the analytical model wouldbe superior in modelling the actual physical response. This is in terms of the equilibriumresponse, the load-carrying capacity and the deformed shape, which is known from earlierresearch to exhibit the change in the local buckling wavelength. The latter point is crucialsince the static FE solution seems to be unable to capture the changing wavelength ofthe post-buckling mode that has been observed in physical experiments. It is likely thatin an actual scenario, the subcritical response in conjunction with the snap-backs arelikely to lead to buckling cells being triggered dynamically. This has been observed duringthe tripping phenomenon (Ronalds, 1989) and the postulate is that the elastic interactivebuckling found currently, if combined with plasticity, would promote this kind of suddenand violent collapse of a stiffened panel.

Currently, the authors are extending the analytical model by including the possibility ofbuckling the main plate in combination with the stiffener. In this case, the load-carryingcapacity is likely to be reduced and it would be representative of a scenario where thecross-section has uniform thickness. Moreover, an imperfection sensitivity study is beingconducted to quantify the parametric space for designers to avoid such dangerous structuralbehaviour.

20

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Becque, J., & Rasmussen, K. J. R. 2009. Experimental investigation of the interaction oflocal and overall buckling of stainless steel I-columns. ASCE J. Struct. Eng., 135(11),1340–1348. DOI: 10.1016/j.jcsr.2009.04.025.

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